Talking about noticing something, do both mean the same?
- I just figured out that the ball is blue.
- I just realized that the ball is blue.
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No. They are substantially different.
To realize, in the sense of coming to a conclusion (and not as in to bring into reality, which is akin to inventing something), means to comprehend something completely.
The act of figuring out is the act of finding a solution to a problem.
Realization usually happens unexpectedly, as if by magic. One minute your mind wanders, and then you see something in a new light. In that sense, it's a passive occurrence.
Figuring something out, on the other hand, takes effort, and is normally done with a specific goal in mind.